How To Write Compound Inequalities

Learning Objectives

Draw sets equally intersections or unions
- Use interval notation to describe intersections and unions
- Use graphs to describe intersections and unions
Solve compound inequalities—OR
- Solve compound inequalities in the course of or and express the solution graphically and with an interval
Solve chemical compound inequalities—AND
- Express solutions to inequalities graphically and with interval notation
- Place solutions for compound inequalities in the form [latex]a<10<b[/latex], including cases with no solution
Solve absolute value inequalities
- Solve single- and multi-step inequalities containing accented values
- Identify cases where there are no solutions to absolute value inequalities

Use interval annotation to depict sets of numbers as intersections and unions

When two inequalities are joined by the word and, the solution of the compound inequality occurs when both inequalities are true at the same time. It is the overlap, or intersection, of the solutions for each inequality. When the two inequalities are joined by the discussion or, the solution of the compound inequality occurs when either of the inequalities is true. The solution is the combination, or union, of the ii individual solutions.

In this section we volition learn how to solve chemical compound inequalities that are joined with the words AND and OR. Get-go, it will help to see some examples of inequalities, intervals, and graphs of chemical compound inequalities. This will help yous describe the solutions to chemical compound inequalities properly.

Venn diagrams utilize the concept of intersections and unions to bear witness how much two or more things share in common. For example, this Venn diagram shows the intersection of people who are breaking your heart and those who are shaking your conviction daily. Manifestly Cecilia has both of these qualities; therefore she is the intersection of the two.

Two circles. One is people who are breaking my heart. The other is people who are shaking my confidence daily. The area where the circles overlap is labeled Cecilia.

In mathematical terms, consider the inequality [latex]x\lt6[/latex] and [latex]x\gt2[/latex]. How would we interpret what numbers x can be, and what would the interval wait like?

In words, x must be less than 6 and at the aforementioned fourth dimension, it must exist greater than 2, much like the Venn diagram above, where Cecilia is at one time breaking your middle and shaking your conviction daily. Allow's look at a graph to see what numbers are possible with these constraints.

x> 2 and x< 6

The numbers that are shared past both lines on the graph are called the intersection of the 2 inequalities [latex]10\lt6[/latex] and [latex]x\gt2[/latex]. This is called a bounded inequality and is written equally [latex]2\lt{10}\lt6[/latex]. Think about that one for a minute. x must be less than six and greater than two—the values for x will autumn between two numbers. In interval notation, this looks like [latex]\left(ii,6\right)[/latex]. The graph would look like this:

Open circle on 2 and open circle on 6 with a line through all numbers between 2 and 6.

On the other paw, if y'all need to stand for two things that don't share whatever common elements or traits, yous can use a wedlock. The following Venn diagram shows two things that share no similar traits or elements but are oftentimes considered in the aforementioned application, such as online shopping or banking.

Two circles, one the Internet and the other Privacy.

In mathematical terms, for example, [latex]x>six[/latex]or [latex]ten<ii[/latex] is an inequality joined by the word or. Using interval notation, we can describe each of these inequalities separately:

[latex]x\gt6[/latex] is the aforementioned as [latex]\left(6, \infty\correct)[/latex] and [latex]ten<2[/latex] is the same as [latex]\left(\infty, 2\right)[/latex]. If we are describing solutions to inequalities, what effect does theorhave? We are maxim that solutions are either real numbers less than 2or real numbers greater than 6. Tin you see why we demand to write them as two carve up intervals? Permit's look at a graph to get a articulate picture of what is going on.

Open circle on 2 and line through all numbers less than 2. Open circle on 6 and line through all numbers grater than 6.

When you lot place both of these inequalities on a graph, we can see that they share no numbers in mutual. This is what we telephone call a union, as mentioned higher up. The interval note associated with a spousal relationship is a large U, so instead of writing or, nosotros join our intervals with a big U, similar this:

[latex]\left(\infty, 2\right)\cup\left(6, \infty\right)[/latex]

Information technology is common convention to construct intervals starting with the value that is furthest left on the number line as the left value, such as [latex]\left(ii,six\right)[/latex], where two is less than 6. The number on the correct should be greater than the number on the left.

Example

Describe the graph of the compound inequality [latex]x\gt3[/latex] or [latex]x\le4[/latex] and describe the set of x-values that volition satisfy information technology with an interval.

In the post-obit video you lot will run into ii examples of how to express inequalities involving OR graphically and as an interval.

Examples

Draw a graph of the chemical compound inequality: [latex]x\lt5[/latex] and [latex]ten\ge−ane[/latex], and describe the gear up of x-values that will satisfy information technology with an interval.

Examples

Describe the graph of the compound inequality [latex]x\lt{-3}[/latex] and [latex]x\gt{iii}[/latex], and depict the set of x-values that volition satisfy it with an interval.

The following video presents two examples of how to draw inequalities involving AND, besides as write the corresponding intervals.

Solve compound inequalities in the class of or

Equally nosotros saw in the last section, the solution of a chemical compound inequality that consists of two inequalities joined with the word or is the spousal relationship of the solutions of each inequality. Unions allow us to create a new set from two that may or may non have elements in common.

In this section you will see that some inequalities demand to be simplified before their solution tin be written or graphed.

In the following example, you volition see an example of how to solve a 1-step inequality in the OR form. Note how each inequality is treated independently until the end where the solution is described in terms of both inequalities. You will use the aforementioned properties to solve compound inequalities that y'all used to solve regular inequalities.

Example

Solve for ten. [latex]3x–i<8[/latex] or [latex]x–five>0[/latex]

Remember to utilize the properties of inequality when you are solving compound inequalities. The side by side instance involves dividing by a negative to isolate a variable.

Case

Solve for y. [latex]2y+7\lt13\text{ or }−3y–ii\lt10[/latex]

In the last case, the final answer included solutions whose intervals overlapped, causing the answer to include all the numbers on the number line. In words, we call this solution "all real numbers." Any existent number will produce a truthful statement for either [latex]y<3\text{ or }y\ge -4[/latex], when it is substituted for ten.

Example

Solve for z. [latex]5z–iii\gt−18[/latex] or [latex]−2z–one\gt15[/latex]

The post-obit video contains an example of solving a compound inequality involving OR, and drawing the associated graph.

In the next section you lot will meet examples of how to solve compound inequalities containing and.

Solve compound inequalities in the form of and and express the solution graphically

The solution of a compound inequality that consists of two inequalities joined with the word and is the intersection of the solutions of each inequality. In other words, both statements must be true at the aforementioned time. The solution to an and chemical compound inequality are all the solutions that the ii inequalities have in mutual. Every bit nosotros saw in the concluding sections, this is where the two graphs overlap.

In this section we will meet more than examples where we have to simplify the compound inequalities before we can limited their solutions graphically or with an interval.

Example

Solve for ten. [latex] \displaystyle 1-4x\le 21\,\,\,\,\text{and}\,\,\,\,5x+2\ge22[/latex]

Instance

Solve for x: [latex] \displaystyle {5}{x}-{2}\le{3}\text{ and }{4}{x}{+vii}>{3}[/latex]

Chemical compound inequalities in the grade [latex]a<x<b[/latex]

Rather than splitting a compound inequality in the form of [latex]a<x<b[/latex] into two inequalities [latex]10<b[/latex] and [latex]x>a[/latex], y'all tin more quickly to solve the inequality past applying the properties of inequality to all iii segments of the compound inequality.

Example

Solve for ten. [latex]3\lt2x+3\leq 7[/latex]

In the video below, you will run across another example of how to solve an inequality in the class [latex]a<10<b[/latex]

To solve inequalities like [latex]a<x<b[/latex], use the addition and multiplication properties of inequality to solve the inequality for x. Whatever operation you perform on the eye portion of the inequality, you must also perform to each of the outside sections also. Pay item attending to division or multiplication past a negative.

The solution to a compound inequality with and is always the overlap between the solution to each inequality. There are three possible outcomes for compound inequalities joined by the give-and-take and:

Case 1:
Description	The solution could be all the values between two endpoints
Inequalities	[latex]x\le{1}[/latex] and [latex]x\gt{-ane}[/latex], or as a bounded inequality: [latex]{-1}\lt{10}\le{one}[/latex]
Interval	[latex]\left(-1,1\correct][/latex]
Graphs
Case 2:
Description	The solution could brainstorm at a point on the number line and extend in ane direction.
Inequalities	[latex]ten\gt3[/latex] and [latex]x\ge4[/latex]
Interval	[latex]\left[4,\infty\correct)[/latex]
Graphs
Case 3:
Clarification	In cases where there is no overlap betwixt the ii inequalities, in that location is no solution to the compound inequality
Inequalities	[latex]ten\lt{-three}[/latex] and [latex]ten\gt{iii}[/latex]
Intervals	[latex]\left(-\infty,-three\right)[/latex] and [latex]\left(3,\infty\right)[/latex]
Graph

In the example below, there is no solution to the compound inequality because there is no overlap between the inequalities.

Instance

Solve for x. [latex]x+2>five[/latex] and [latex]x+four<five[/latex]

Solve inequalities containing absolute values

Let's apply what you know well-nigh solving equations that contain absolute values and what y'all know about inequalities to solve inequalities that contain absolute values. Let's showtime with a simple inequality.

[latex]\left|x\right|\leq 4[/latex]

This inequality is read, "the absolute value of 10 is less than or equal to iv." If yous are asked to solve for 10, y'all want to find out what values of x are iv units or less away from 0 on a number line. You could outset by thinking about the number line and what values of ten would satisfy this equation.

4 and [latex]−iv[/latex] are both four units abroad from 0, and so they are solutions. 3 and [latex]−3[/latex] are also solutions because each of these values is less than 4 units away from 0. And so are one and [latex]−1[/latex], 0.5 and [latex]−0.5[/latex], and so on—there are an space number of values for x that volition satisfy this inequality.

The graph of this inequality volition have two airtight circles, at four and [latex]−4[/latex]. The distance between these two values on the number line is colored in blue because all of these values satisfy the inequality.

Number line. Closed blue circles on negative 4 and 4. Blue line between closed blue circles.

The solution can exist written this way:

Inequality: [latex]-4\leq x\leq4[/latex]

Interval: [latex]\left[-iv,iv\right][/latex]

The situation is a little unlike when the inequality sign is "greater than" or "greater than or equal to." Consider the elementary inequality [latex]\left|x\correct|>3[/latex]. Over again, you could remember of the number line and what values of x are greater than 3 units away from zero. This fourth dimension, three and [latex]−iii[/latex] are non included in the solution, so there are open circles on both of these values. two and [latex]−ii[/latex] would not be solutions because they are not more than 3 units away from 0. But five and [latex]−five[/latex] would work, and then would all of the values extending to the left of [latex]−3[/latex] and to the right of iii. The graph would look like the one below.

Number line. Open blue circles on negative three and three. Blue arrow through all numbers less than negative 3. Blue arrow through all numbers greater than 3.

The solution to this inequality tin can be written this fashion:

Inequality: [latex]x<−three[/latex] or [latex]x>iii[/latex].

Interval: [latex]\left(-\infty, -iii\right)\cup\left(three,\infty\correct)[/latex]

In the post-obit video, you volition see examples of how to solve and limited the solution to absolute value inequalities involving both AND and OR.

Writing Solutions to Accented Value Inequalities

For any positive value of aandx, a single variable, or any algebraic expression:

Absolute Value Inequality	Equivalent Inequality	Interval Notation
[latex]\left\|{ x }\correct\|\le{ a}[/latex]	[latex]{ -a}\le{x}\le{ a}[/latex]	[latex]\left[-a, a\right][/latex]
[latex]\left\| ten \right\|\lt{a}[/latex]	[latex]{ -a}\lt{x}\lt{ a}[/latex]	[latex]\left(-a, a\correct)[/latex]
[latex]\left\| x \correct\|\ge{ a}[/latex]	[latex]{x}\le\text{−a}[/latex] or [latex]{10}\ge{ a}[/latex]	[latex]\left(-\infty,-a\right]\loving cup\left[a,\infty\right)[/latex]
[latex]\left\| x \right\|\gt\text{a}[/latex]	[latex]\displaystyle{x}\lt\text{−a}[/latex] or [latex]{10}\gt{ a}[/latex]	[latex]\left(-\infty,-a\right)\cup\left(a,\infty\right)[/latex]

Permit's look at a few more examples of inequalities containing absolute values.

Instance

Solve for x. [latex]\left|ten+3\right|\gt4[/latex]

Example

Solve for y.[latex] \displaystyle \mathsf{3}\left| \mathsf{2}\mathrm{y}\mathsf{+6} \right|-\mathsf{ix<27}[/latex]

In the post-obit video, you will see an example of solving multi-step absolute value inequalities involving an OR situation.

In the following video you will see an example of solving multi-footstep absolute value inequalities involving an AND situation.

In the last video that follows, you will see an example of solving an absolute value inequality where yous demand to isolate the accented value starting time.

Identify cases of inequalities containing absolute values that have no solutions

As with equations, there may exist instances in which there is no solution to an inequality.

Instance

Solve for ten. [latex]\left|2x+iii\right|+nine\leq 7[/latex]

Summary

A compound inequality is a statement of two inequality statements linked together either by the discussion or or by the word and. Sometimes, an and compound inequality is shown symbolically, like [latex]a<x<b[/latex], and does not even demand the word and. Because compound inequalities stand for either a wedlock or intersection of the private inequalities, graphing them on a number line can be a helpful way to see or bank check a solution. Compound inequalities can be manipulated and solved in much the aforementioned manner any inequality is solved, by paying attention to the properties of inequalities and the rules for solving them.

Absolute inequalities can exist solved by rewriting them using chemical compound inequalities. The first step to solving absolute inequalities is to isolate the accented value. The next step is to determine whether you are working with an OR inequality or an AND inequality. If the inequality is greater than a number, we volition use OR. If the inequality is less than a number, we will utilise AND. Think that if we end up with an accented value greater than or less than a negative number, in that location is no solution.